Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{6a^3 + 48a^2 - 54a}{10a^2 + 160a + 630}$
First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {6a(a^2 + 8a - 9)} {10(a^2 + 16a + 63)} $ $ z = \dfrac{6a}{10} \cdot \dfrac{a^2 + 8a - 9}{a^2 + 16a + 63} $ Simplify: $ z = \dfrac{3a}{5} \cdot \dfrac{a^2 + 8a - 9}{a^2 + 16a + 63}$ Next factor the numerator and denominator. $ z = \dfrac{3a}{5} \cdot \dfrac{(a + 9)(a - 1)}{(a + 9)(a + 7)}$ Assuming $a \neq -9$ , we can cancel the $a + 9$ $ z = \dfrac{3a}{5} \cdot \dfrac{a - 1}{a + 7}$ Therefore: $ z = \dfrac{ 3a(a - 1)}{ 5(a + 7)}$, $a \neq -9$